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Friezes and Mosaics
The Mathematics of Beauty
Frieze pattern on the walls of theTaj
Mahal.
Frieze patterns in the Taj
Mahal
The gardens and corridors have many frieze patterns.
Mosaics in the Taj
Mahal
The ground around the Taj
Mahal
is
laid with a tiling pattern of four-
pointed stars.
Palace of mirrors in Jaipur, India
The palace complex at Amer
Fort near Jaipur has a hall of
mirrors.
During the day, the chamber reflectssunlight and at night, a single candleis reflected multiple times enough toilluminate the room.
Jaisalmer
in Rajasthan
These are frieze patterns appearing on the walls of the Jaisalmer
Fort in
Rajasthan, India.
Friezes
We will look at the symmetries of these seven frieze patterns.
Simplified friezes
These exhibit translational, rotational and reflective symmetries.
Main theorem for symmetry groups of friezes
There are only 7 possible symmetry groups for any frieze pattern.
They are listed as: (1) <tL
>, group generated by a translation of length L.
(2) <tL
, rv
>, with vertical reflection rv
.
(3) <tL
, rh
>, with horizontal reflection rh
.
(4) <tL
, tL/2
rh
>.
(5) <tL
, rh
rv
>.
(6) <tL
, tL/2
rh , rh
rv
>.
(7) <tL
, rh
, rv
>.
Mosaics
A mosaic is a pattern that can be repeated to fill the plane and it is periodic along two independent directions.
Main theorem for symmetry groups of mosaics
There are only 17 symmetry groups and these can be listed.
The simplest is the group generated by a single translation.(p1)
The groups pg and pm
The group pg contains glide reflections only and their axes are parallel.
The group pm has no rotations and only reflection axes which are parallel.
cm, p2 and pgg
pmg, pmm
and cmm
p3, p31m, and p3m1
There are five more crystallographic groups: p4, p4g, p4m, p6 and p6m.
Five more …